Spinning up a progressively smaller sphere

[csmcmillion]

I was watching a physics lecture and the instructor made the statement that "for a given mass, decreasing the diameter of the body makes it harder to spin up". In one way this makes sense, as with a smaller diameter one has a shorter torque arm to work with... however, I seem to recall from my physics that the moment of inertia goes as r^2. Which would say that it would be easier to spin up as the size decreases.

What am I missing?

Thanks!

 

[gsal]

I think you are right.

Are you sure he did not say "makes it spin up harder", instead? That would make a lot of sense, that's what ice skater do...they have constant mass but when they reduce their radius (by bringing arms to the chest) they spin faster given the same energy...that means, it became easier to spin.

 

[csmcmillion]

I think you are right.

Are you sure he did not say "makes it spin up harder", instead?

I'm sure. This was in a QM context and he was talking about how hard it is to give an electron a quantum of spin due to it's very small size.

 

[gsal]

Well...if it is THAT small, no wonder it is difficult...try to spin something with radius zero!

Nevertheless, (for a given force) torque will go down proportional to radius and moment of inertia will go down proportional to the square of the radius...so, I still think things get easier and easier.

If the radius goes down from 5 to 4, the torque decreases 20% but the moment of inertia went down from k25 to k16, that's 36%

no?

 

[csmcmillion]

Well...if it is THAT small, no wonder it is difficult...try to spin something with radius zero!

Nevertheless, (for a given force) torque will go down proportional to radius and moment of inertia will go down proportional to the square of the radius...so, I still think things get easier and easier.

If the radius goes down from 5 to 4, the torque decreases 20% but the moment of inertia went down from k25 to k16, that's 36%

no?

Right but he (Susskind) explicitly says the Electron has a finite radius. I agree with your calculations. I'm still scratching my head over this.

 

[chrisbaird]

I think your instructor is wrong or you misunderstood him. It would be easier to spin up a smaller object with same mass. Think about it, which one has more energy, two buckets of water tied together by a foot long rope and spun around the system's center of mass, or the same buckets tied together by a 10 foot rope and spun at the same speed? (If you cut the rope while spinning, which will make a bigger splash?) To start from rest and spin up the larger system would take more energy, and thus is "harder" (whatever that means).

 

[csmcmillion]

I think your instructor is wrong or you misunderstood him.

I'm sure I didn't misunderstand, as he stated it in two different (video) lectures. He could be wrong, but he (Susskind) is no lightweight.